Stochastic processes
Stochastic processes
Fundamentals of Signals & Transmissions
This MOOC is part of the Fundamentals of Signals & Transmissions series, which introduces the core principles of signals, probability and stochastic processes, providing essential tools to understand, analyze and design modern communication systems.
See the full seriesCourse description
The MOOC offers a practical knowledge on stochastic processes, providing tools to analyse random signals understand and predict, in statistical terms, their behaviour in time or space.
Beginning with the characterization of stationary, ergodic processes, the course delves into correlation and spectra, discussing their estimate and use. It shows how linear, time-invariant systems responds to random signals and how input / output statistics captures their behaviour.
The theoretical content is complemented by simple examples of data analytics in MATLAB.
Total workload of the course: 13 hours
This MOOC is provided by Politecnico di Milano.
This MOOC was produced as part of the Edvance project – Digital Education Hub per la Cultura Digitale Avanzata. The project is funded by the European Union – Next Generation EU, Component 1, Investment 3.4 “Didattica e competenze universitarie avanzate".
Intended Learning Outcomes
At the end of this course, you will be able to:
- Classify stochastic processes as continuous or discrete, and as monodimensional or multidimensional, and describe their characterization in terms of probabilities and statistical moments (mean, standard deviation, cross-correlation).
- Analyze the properties of stationary and ergodic processes and apply the concept of ergodicity to extract statistical information.
- Explain how the power spectrum characterizes stochastic processes, relate it to autocorrelation, and evaluate its significance in practical scenarios.
- Determine the relationship between the input and output of stochastic processes in a Linear Time-Invariant (LTI) system.
- Assess the relevance of stochastic processes, particularly Gaussian processes, and illustrate their application in real-world contexts.
ESCO: statistics ESCO: probability theory ESCO: signal processing
Prerequisites
A foundational knowledge of signals and systems, along with basic probability concepts, is expected. Completing the other MOOCs in the 'Fundamentals of Signals & Transmissions' series is recommended to strengthen your preparation.
Activities
Over and above consulting the content, in the form of videos and other web-based resources, you will have the opportunity to discuss course topics and to share ideas with your peers in the Forum of this MOOC. The forum of this MOOC is freely accessible, and participation is not guided; you can use it to compare yourself with other participants, or to discuss course contents with them.
Section outline
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The Week consists of the following lessons:
Introduction to stochastic processes. We show what a stochastic process is, when it is stationary, and how it can be modelled and characterized by first and higher order statistics, and we discuss the role of moments like mean and cross-correlation.
Stationarity, ergodicity, and memory. The concept of ergodicity allows us to link temporal and ensemble statistics and estimate statistics by a time-limited observation. We also show how to derive statistics of discrete processes from continuous ones.
Power Spectrum Density is introduced, illustrating its link with autocorrelation, uses, and estimation: the periodogram (benefits and limitations) and the spectrum analyzer. White processes are defined in continuous and discrete cases.
Processes and LTI systems. The lesson shows how Linear Time Invariant interacts with processes: how input and output statistics like cross-spectra or auto-correlation are related to the LTI impulse response, and how these properties can be used in practical applications.
Examples and applications. Three examples from real-world applications are introduced and discussed within two talks.
The first is estimating delay at the base of positioning and navigation in GNSS or RADAR. It is carried out as a complete analysis, showing how to assess and qualify the delay in a simple but instructive case.
The second example illustrates the base of the numerical transmissions, and one application to transmission over huge (astronomic) distances.
The third and last example refers to the principle of operation of a linear antenna array, and how the data collected by the elementary antennas can be processed to digitally ‘point’ the array.
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Video transcripts Folder
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Bibliography and Linkography Page
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Lesson slides Folder
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Assessment
Your final grade for the course will be based on the results of your answers to the assessed quizzes. You have an unlimited number of attempts at each quiz, but you must wait 15 minutes before you can try again. You will have successfully completed the course if you score 60% (or higher) in each one of the assessed quizzes. The maximum score possible for each quiz is given at the beginning of the quiz. You can view your score in the quiz on your last attempt or on the 'Grades' page.
Certificate
You can achieve a certificate in the form of an Open Badge for this course, if you reach at least 60% of the total score in each one of the assessed quizzes and fill in the final survey.
Once you have completed the required tasks, you will be able to access ‘Get the Open Badge’ and start issuing the badge. Instructions on how to access the badge will be sent to your e-mail address.
The Badge does not confer any academic credit, grade or degree.
Information about fees and access to materials
The course is delivered in online mode and is available free of charge.
Course faculty
Andrea Virgilio Monti-Guarnieri
Teacher
Andrea Virgilio Monti-Guarnieri, IEEE senior member, M. Sc cum laude (1988) in electronic engineering, IEEE senior member, full professor within “Dipartimento di Elettronica, Informazione e Bioingegneria” (2017). H index (Google Scholar): 43, 8800 citations, five conference awards, and five patent applications. He has been a reviewer and editor of several scientific journals and a member of scientific-technical committees of international workshops and symposia on Radar and Earth Observation (EO).
He co-founded Aresys (2003), targeting SAR and Radar-based applications and customized solutions. He is a past member of the Technical-scientific Committee of the Italian Space Agency (ASI), national delegate of the Group on Earth Observations (GEO), and European Space Agency Mission Advisory Group for Hydroterra.
He has long-standing experience in Spaceborne Synthetic Aperture Radar. He participated in the design, calibration, and quality assessment of Italian, European, and Argentinean Spaceborne SAR missions, cooperating with the national and European Space Agencies for 30 years.
His interests focus on cutting-edge radar technologies, including MIMO LEO SAR formations, ground-based and satellite radar systems for environmental monitoring, civil, and security applications. He is presently the prime investigator of the geostationary SAR mission Hydroterra+, candidate Earth Explorer-12 and of Italian Space Agency Mission Advisory Group for GEO-SAR mission.
Contact details
If you have any enquiries about the course or if you need technical assistance please contact pok@polimi.it. For further information, see FAQ page.